Vick threw for 237 yards, LeSean McCoy ran for 126 more and Philadelphia safety Kurt Coleman recorded three of the Eagles' four interceptions.
Philadelphia improved to 2-4 and the Redskins slipped out of first place in the NFC East by falling to 3-2. The New York Giants (4-2) moved past Washington into the division lead with their win over Buffalo.
Beck led Washington on an 80-yard touchdown drive in the fourth quarter and scored on a 2-yard run to bring the Redskins with 20-13.with 2:44 to play.
The Redskins chose to kick deep after the touchdown and they never got the ball back. Philadelphia picked up a first down on 3rd-and-inches and then ran out the clock.
Washington kept Philadelphia from scoring in the second half, but the interceptions cost the Redskins a realistic chance to win the game.
The Eagles scored 10 points from two of the interceptions while dominating the first half. Philadelphia had 16 first downs and kept the ball for more than 21 minutes during the first two quarters, outgaining Washington 250 yards to 75.
Vick's touchdown pass was a 7-yarder to Brent Celek midway through the first period. McCoy scored on a 1-yard run early in the second and Alex Henery added field goals of 28 and 24 yards for the Eagles as they built their 20-0 lead.